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Lagrangian solutions for the semi-geostrophic shallow water system in physical space with general initial data

Authors: M. Feldman and A. Tudorascu
Original publication: Algebra i Analiz, tom 27 (2015), nomer 3.
Journal: St. Petersburg Math. J. 27 (2016), 547-568
MSC (2010): Primary 76U05
Published electronically: March 30, 2016
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Abstract | References | Similar Articles | Additional Information

Abstract: In order to accommodate general initial data, an appropriately relaxed notion of renormalized Lagrangian solutions for the Semi-Geostrophic Shallow Water system in physical space is introduced. This is shown to be consistent with previous notions, generalizing them. A weak stability result is obtained first, followed by a general existence result whose proof employs the said stability and approximating solutions with regular initial data. The renormalization property ensures the return from physical to dual space and ultimately enables us to achieve the desired results.

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  • 1. L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004), no. 2, 227-260. MR 2096794
  • 2. L. Ambrosio, M. Colombo, G. De Philippis, and A. Figalli, Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case, Comm. Partial Differential Equations 37 (2012), no. 12, 2209-2227. MR 3005541
  • 3. -, A global existence result for the semigeostrophic equations in three dimensional convex domains, Preprint, arXiv:1205.5435, 2012.
  • 4. L. Ambrosio and W. Gangbo, Hamiltonian ODE in the Wasserstein spaces of probability measures, Comm. Pure Appl. Math. 61 (2008), no. 1, 18-53. MR 2361303
  • 5. L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and the Wasserstein spaces of probability measures, Lectures Math. ETH Zurich, Birkhäuser, Basel, 2005. MR 2129498
  • 6. J.-D. Benamou and Y. Brenier, Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampere/transport problem, SIAM J. Appl. Math. 58 (1998), no. 5, 1450-1461. MR 1627555
  • 7. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375-417. MR 1100809
  • 8. M. Cullen, Private communication.
  • 9. M. Cullen and M. Feldman, Lagrangian solutions of semigeostrophic equations in physical space, SIAM J. Math. Anal. 37 (2006), no. 5, 1371-1395. MR 2215268
  • 10. M. Cullen and W. Gangbo, A variational approach for the 2-dimensional semi-geostrophic shallow water equations, Arch. Ration. Mech. Anal. 156 (2001), 241-273. MR 1816477
  • 11. M. Cullen and H. Maroofi, The fully compressible semi-geostrophic system from meteorology, Arch. Ration. Mech. Anal. 167 (2003), no. 4, 309-336. MR 1981860
  • 12. M. Cullen and R. J. Purser, An extended Lagrangian theory of semi-geostrophic frontogenesis, J. Atmospheric Sci. 41 (1984), no. 9, 1477-1497. MR 881109
  • 13. L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001)
  • 14. J. C. O. Faria, F. Lopes, and H. J. Nussenzveig, Weak stability of Lagrangian solutions to the semigeostrophic equations, Nonlinearity 22 (2009), 2521-2539. MR 2539766
  • 15. M. Feldman and A. Tudorascu, On Lagrangian solutions for the semigeostrophic system with singular initial data, SIAM J. Math. Anal. 45 (2013), no. 3, 1616-1640. MR 3061466
  • 16. -, On the semigeostrophic system in physical space with general initial data. (to appear)
  • 17. W. Gangbo, T. Nguyen, and A. Tudorascu, Euler-Poisson systems as action-minimizing paths in the Wasserstein space, Arch. Ration. Mech. Anal. 192 (2009), no. 3, 419-452. MR 2505360
  • 18. B. Hoskins, The geostrophic momentum approximation and the semigeostrophic equations, J. Atmospheric. Sci. 32 (1975), no. 2, 233-242.
  • 19. C. Villani, Topics in optimal transportation, Grad. Stud. Math., vol. 58, Amer. Math. Soc., Providence, RI, 2003. MR 1964483

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Additional Information

M. Feldman
Affiliation: Department of Mathematics, University of Wisconsin–Madison, Madison, Wisconsin 53706

A. Tudorascu
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506

Keywords: Semi-Geostrophic Shallow Water system, flows of maps, optimal mass transport, Wasserstein metric, optimal maps, absolutely continuous curves
Received by editor(s): November 25, 2014
Published electronically: March 30, 2016
Additional Notes: The authors would like to thank M. Cullen for his valuable suggestions and comments. The work of Mikhail Feldman was supported in part by the National Science Foundation under Grant DMS-1101260, and by the Simons Foundation under the Simons Fellows program. This work was partially supported by a grant from the Simons Foundation (#246063 to Adrian Tudorascu)
Dedicated: Dedicated to Nina Nikolaevna Ural’tseva on her 80th birthday
Article copyright: © Copyright 2016 American Mathematical Society

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